Steinberg group (K-theory)

In algebraic K-theory, a field of mathematics, the Steinberg group $\mathrm{St}(A)$ of a ring $A$ is the universal central extension of the commutator subgroup of the stable general linear group of $A$.

It is named after Robert Steinberg, and it is connected with lower $K$-groups, notably $K_2$ and $K_3$.

Definition

Abstractly, given a ring $A$, the Steinberg group $\mathrm{St}(A)$ is the universal central extension of the commutator subgroup of the stable general linear group (the commutator subgroup is perfect and so has a universal central extension).

Presentation using generators and relations

A concrete presentation using generators and relations is as follows. Elementary matrices — i.e. matrices of the form $e_{pq}(\lambda ):=\mathrm{𝟏}+a_{pq}(\lambda )$, where $\mathrm{𝟏}$ is the identity matrix, $a_{pq}(\lambda )$ is the matrix with $\lambda$ in the $(p,q)$-entry and zeros elsewhere, and $p\ne q$ — satisfy the following relations, called the Steinberg relations:

$\begin{array}{cccc}{e}_{ij}(\lambda )e_{ij}(\mu )& =e_{ij}(\lambda +\mu );& & \\ [e_{ij}(\lambda ),e_{jk}(\mu )]& =e_{ik}(\lambda \mu ),& & \text{for }i\ne k;\\ [e_{ij}(\lambda ),e_{kl}(\mu )]& =\mathrm{𝟏},& & \text{for }i\ne l\text{ and }j\ne k.\end{array}$

The unstable Steinberg group of order $r$ over $A$, denoted by ${\mathrm{St}}_r(A)$, is defined by the generators $x_{ij}(\lambda )$, where $1\le i\ne j\le r$ and $\lambda \in A$, these generators being subject to the Steinberg relations. The stable Steinberg group, denoted by $\mathrm{St}(A)$, is the direct limit of the system ${\mathrm{St}}_r(A)\to {\mathrm{St}}_{r+1}(A)$. It can also be thought of as the Steinberg group of infinite order.

Mapping $x_{ij}(\lambda )\mapsto e_{ij}(\lambda )$ yields a group homomorphism $\phi :\mathrm{St}(A)\to {\mathrm{GL}}_{\mathrm{\infty }}(A)$. As the elementary matrices generate the commutator subgroup, this mapping is surjective onto the commutator subgroup.

Interpretation as a fundamental group

The Steinberg group is the fundamental group of the Volodin space, which is the union of classifying spaces of the unipotent subgroups of $\mathrm{GL}(A)$.

Relation to K-theory

K₁

$K_1(A)$ is the cokernel of the map $\phi :\mathrm{St}(A)\to {\mathrm{GL}}_{\mathrm{\infty }}(A)$, as $K_1$ is the abelianization of ${\mathrm{GL}}_{\mathrm{\infty }}(A)$ and the mapping $\phi$ is surjective onto the commutator subgroup.

K₂

$K_2(A)$ is the center of the Steinberg group. This was Milnor's definition, and it also follows from more general definitions of higher $K$-groups.

It is also the kernel of the mapping $\phi :\mathrm{St}(A)\to {\mathrm{GL}}_{\mathrm{\infty }}(A)$. Indeed, there is an exact sequence

$1\to K_2(A)\to \mathrm{St}(A)\to {\mathrm{GL}}_{\mathrm{\infty }}(A)\to K_1(A)\to 1.$

Equivalently, it is the Schur multiplier of the group of elementary matrices, so it is also a homology group: $K_2(A)=H_2(E(A);\mathbb{Z})$.

K₃

Template:Harvtxt showed that $K_3(A)=H_3(\mathrm{St}(A);\mathbb{Z})$.

References

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